the behavior of graphite under biaxial …academic.csuohio.edu/duffy_s/temp/doe caes program/stress...

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL TENSION*

W. L. GREENSTREET, G. T. YAHR and R. S. VALACHOVIC Oak Ridge National Laboratory, Oak Ridge, I‘ennessee 37830, I,! .S..I.

(Received I SJune 1972)

Abstract- A small-deformation elastic-plastic continuum theory for describing the stress-

strain behavior of graphite is examined for specimens under biaxial loading at room temperature. The continuum theory takes into account the anisotropy and plastic com- pressibility of graphite as well as the continuous nonlinearity of the stress-strain response. The measured strains for thin-walled cylindrical shells loaded by internal pressure arc shown to agree \‘ery closely with the predictions of the theory for both loading and unloading.

1. INTRODUCTION

This paper describes the examination of an elastic-plastic continuum theory which was developed to describe the nonlinear mechanical behavior of artificial graphite. The basis of this examination is a comparison between calculated and experimental results for combined stress conditions. Thin-walled cylinders made from extruded graphite were internally pressurized to provide data on loading and unloading response, and the experimentally obtained stress-strain curves were compared with calculated results.

In the main, the circular cylindrical specimens were made from extruded Graphitite-(; tubing. Five specimens of’ this material were tested, while a single specimen, which was made from a specialty graphite, was also used. In each case, the material exhibited transversely isotropic behavior. with the axis of. isotropic symmetry being in the direction of’the longitudinal axis of the specimen.

The theory examined is briefly described in the next section. The equations employed in this stud), were based on approximate,

Wesearch sponsored by the U.S. Atomic Etnergy Commission under contract with the Union Car- bide Corporation.

but mathematically the simplest, analogs to the physical behavior postulated in the de\,elopment of the theory.

The second section describes the tests performed using specimens made from Graphitite-G material. Because of’ \,ariations in properties associated with this material, special procedures for data collection were used to overcome masking effects caused by \,ariations in properties. The use of these procedures allowed parameters in the constitutive equations to be determined t‘rom a combination of’ uniaxial test results and results obtained directly from the specimens af’ter the combined stress tests were com- pleted.

The third section describes the Graphitite- G test results. Further discussion of’ the

elastic-plastic continuum theory is also given in this section. The f’ourth section gives comparisons batween calculated and experimental results for the Graphitite-(; and the specialty graphite specimens.

2. MATHEMATICAL THEORY

The small-deformation elastic-plastic

continuum theory [ l-31, which was developed to describe artificial graphite beha\%~r, is summarized, and the stress-strain relations

43

44 W. L. GREENSTREET, G. T. YAHR and R. S. VALACHOVIC

for pressurized thin-walled cylinders are given in this section. In the theory, as in the classical mathematical theory of plasticity, it is postulated that the total symmetric strain tensor, l KL, can be decomposed into elastic and plastic components, that is,

EKL = E;L -+ c;L, W,L= 1,%3) (1)

where l iL are the elastic components and l PKL are the plastic components. The elastic strains are given by the generalized Hooke’s law

EiL = SKLMN CMN, (2)

Equation (4) describes the surface associated with initial or first loading from the stress-free state of a virgin material. Loading reversal. occurs when the stress point, which represents the stress state in stress space, moves toward the interior of the current loading surface. As a result of loading reversal, a new surface is generated.

Consider the case in which the stress point has moved along an initial loading path in stress space until a stress state denoted by oKL = ugL is reached. A reversal of loading at WKL = m;L will start generating a loading surface given by

where SKLMN = SMNKL represents the elastic & = hKLMN (TKL,ll~,\lN,,, = K(lh (5) compliances and oKL = (TLK is the stress tensor. Equation (2) can be written in incre-

where

mental form as o KL,,, = (TKL - fl$L (6)

dGL = SKLMN dg,N. (3)

Prior to discussing the stress-plastic strain relations, loading, or potential, surface behavior must be considered. In the case of artificial graphite, a yield surface in the classical sense does not exist; that is, there exists no surface which encloses a region of purely elastic response. However, the concept of a loading surface, distinct from a yield surface, can be introduced. As shown in Refs. [l] and [2], the loading surface, or plastic potential surface, for artificial graphite may be expressed by

f = &KLMflKLflMN = K, (4)

where the constants aKLMN satisfy the relations aKLMN = aMNKL and K is a scalar function which depends on the history of loading. Equation (4) represents the simplest mathematical form for describing loading surfaces appropriate to the behavior of graphite. For a more general expression for-f, which allows for translation and growth of the loading surfaces, the reader is referred to Refs. [l] and [2].

Here ogL represents a pseudovirgin state, and the stress during loading reversal is measured from azL in the same way as it is measured from the zero state during the first loading of the virgin body.

A new reversal of loading at cr* will again start generating a new loading surface, which is given in this case by

where

c KL,l, = uKL - dL,,, . (8)

This process is repeated for every loading reversal.

Before proceeding with the discussion of the elastic-plastic theory, we shall consider uniaxial loading. In his studies of uniaxial loading ofgraphite, Jenkins [4] tacitly assumed the validity of the postulate expressed by equation (1) and established equations of the following forms for describing stress vs. longitudinal strain diagrams from uniaxial tests:

E = AuS-Bu2; (9)

for initial loading,

E-Cm = A(rr-a,J -cB(a-urn)2 (10)

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL TENSION 45

d8 (‘3)

Similar expressions apply for successive loading reversals. In equation (IS), (’ is the same as the constant c in equations (10) and (11). Further,

for unloading from a maximum stress cm with the corresponding strain E,, and?

E-~,,=A(tr-u~)-tC~(u-u(,)2 (11)

for reloading from (T,, with the coresponding strain E,, when CT c Urn. In equation (9), (lo), and (ll), A (= l/E) is the elastic com- pliance, H is a material constant which charac- terizes the plastic deformation, and c is a constant. E is the elastic modulus.

-Jenkins’ development [4] resulted in a value of one-half for c, with the value being independent of orientation with respect to the grain, that is, independent of whether the specimen is oriented in the against-grain or with-grain direction. In addition, this constant does not depend upon whether the specimen is being unloaded or reloaded. Since his equations were shown[4,6-81 to give good to exceptional representations of graphite behavior, the plastic strains are taken as quadratic functions of the stresses in the mathematical analog to elastic-plastic behavior used here. However, the results of our investigations show that c does not neces- sarily take the value of one-half as implied by Jenkins, and it is for this reason that we have included it as an unknown constant in equations (10) and (11).

Following the developments in Refs. [I] and [2], the relation between plastic strain increments and stress increments on first loading is given by

The relation for the first loading reversal is

cLL,,, = & - l A-l,, (14)

where c;;, is the plastic strain existing at the instant of reversal at a;,,

The relation between total strain increments and stress increments on first loading is

(1.5)

For the first loading reversal the relation is

af‘ + [?f;:,]l~2 aa

a/’ dtr - acrnr,,, MI,,, .

(16) Here,

de KI.,,, = de,, - de;,.,

where de:,. is the total strain at the instant of loading reversal at r&.

To simplify the ensuing discussion, we use contracted notation in which the stress and strain notations have the following equivalences:

and

tin his first paper[4], Jenkins took the specific case (TV = 0. In a later paper [5], illustrations of stress-strain curves are given for stresses both in tension and comoression.

61 = El1 E‘j = 2E 23 = y23

Ep = $2 E:, = 2E3, = y:<1 (18)

E3 = %:3 Eg = 2E 12 = YIZ.

Note that hK,,(K # L) are the engineering shear strain components.


As mentioned in the Introduction, the material is assumed to be traversely isotropic, and, throughout the subsequent discussions, the 3-axis is the axis of anisotropic symmetry and the I- and P-axes lie within the plane of symmetry,f In contracted notation the constitutive equation, equation (15), becomes

(K, L = 1,2,3,4,5, or 6) (19)

and the plastic potential equation, equation (4), becomes

(20)

As stated above, the compliances and the coefficients, or constants, in the plastic potential equation are symmetrical. In contracted notations the symmetry relations become

SKI, = SLK

and

a,& = aLn_-

We are now in a position to discuss the constants and their relationships to quantities determined from uniaxial tests. The nonzero compliances and their relationships to the usual elastic constants for a transversely isotropic material aref

S12 zzc s21 = --y12 -_ -- VZI

EII En’

s23 = $32 = si3 = f3l - V13 _ -___-- E

I1 “.“,‘,s c2.f) F.

tl and 2 are against-grain directions, while 3 is the with-grain direction.

$See Love [9] or Hearmon [IO].

1 s33 =-f

E33

where

E = Young’s modulus, G = elastic shear modulus,

VMN = Poisson’s ratio; the first subscript indicates the direction of applied stress and the second indicates the direction of induced strain.

Note that the constant A in equations (9), (lo), and (11) corresponds to sll (= sZ2) and s33.

Through the use of Ref. [l], the following equations can be written for relating the nonzero agL to parameters that can be determined from uniaxial test results:

alI = az2 = (2B11)2’3,

a12 = azl = --kc2 mw2'3,

a23 = al3 = a31 = a32 = --pT3 (2B,,)2’3

a33 = (~BxJ~'~,

= -& (2B~3)~‘~, (22)

a44 = a55 = (M23Yf3,

Nere, Btl(=B,,) is the B value [equation (911 for the against-grain direction, BQ3 is Ihe B value for the with-grain direction, and B23

is determined from a shear stress versus shear strain diagram. The &, pT3, and p& are the plastic strain ratios. [Again, the subscripts on pgL (K rk L) have the same meanings as those for VKL (K # L).]

The constitutive equations giving the axial and circumferential strains are written in expanded form below. These equations will be used in a subsequent section for making comparisons between predictions of this theory and measured strains from the thin-

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL. TENSION 17

walled cylinders. For initial loading,

and

(23)

(24)

where l q is the circumferential strain, E:~ is the axial strain, and

R = 2 = constant L

is the axial-to-circumferential stress ratio. The corresponding equations for first load reversal ‘Jr unloading are

and

t%,, = s33(R - v:,,)n2,,>

(25)

The quantities Q,,, c3,>,, UP,,,, and u3,,,

[= Ku,, I ) 1 are given by

E 2,,, = E2 - E$, l . J,,) = Es-E;,

‘J “,,, = <r2- cr$, (T ‘\,,> = (Tg- (T::.

The stresses a$ and a; are those associated with the load reversal or unloading point; the strains E._? and l 2 are those that existed

at the instant of load reversal. Note that the starred quantities were designated by E,, and crm in equation (10).

3. TESTS AND TESTING PROCEDURES, GRAPHITITE-G MATERIAL

In this section we describe the biaxial tests conducted on the thin-walled cylindrical specimens as well as uniaxial tests performed to obtain material property data. The material property data were obtained from uniaxial tests on the cylindrical specimens following acquisition of the biaxial data and from standard tensile specimens. These data were used to determine the parameters for use in equations (23) through (26).

Tests were conducted on five cylindrical specimens of the design shown in Fig. 1. As may be seen from the figure, the gage sections of the cylindrical specimens were 6-in. long, with a 0.092-in. wall thickness, and a mean radius of 0.922 in. The specimens were fabricated from tubes having a liin. inside diameter and a 2 in. outside diameter. Short cylindrical sleeves were machined from 2-in.-inside diameter by 2$in.-outside diameter tube stock and were glued over each end of IO-in.-long cylinders, which were machined to form the gage sections. ~l-he composite structures were then machined to the final specimen configuration. Since the specimens were parallel with the with- grain direction of the graphite; the material was isotropic in planes normal to the axes of the cylinders.

Each cylindrical specimen was instrumented on the outside surface with strain gages oriented in the axial and circumferential directions. Type C6-121-A Budd Metalfilm gages with a 0.125-in. gage length were used, and they were mounted in pairs (one axial gage and one circumferential gage) at selected locations. One specimen (no. 1) had two sets mounted 180-deg apart at the center of the gage section, while the remaining four had four pairs mounted at SO-deg intervals around the circumference at the


DRNL_DWG 69_5046 force only. A specimen mounted in a loading

Fig. 1. Design of thin-walled cylindrical specimen.

center of the gage section. In addition, pairs of gages were mounted in the transition regions at the ends of each specimen. In two cases, the specimens (nos. 2 and 3) were extensively strain gaged to obtain strain distributions as functions of axial position.

Figure 2 shows a cylindrical specimen mounted in a loading fixture for applying internal pressure only or an axial tensile

fixture for applying internal pressure plus a compressive force so that no net axial load

[~ t is exerted on the specimen is shown in Fig. 3. ‘? Each specimen was subjected to biaxial ! 2~49 loading; following this, uniaxial loading 1 I conditions were imposed in various sequences.

A summary of the tests and test conditions is given in Table 1. The loadings used and the sequences of applications are given in column 2 of the table, where the circumferential to axial stress ratio for each loading is also listed, A cycle consisted of loading to the maximum pressure or load and unloading to zero pressure or load. The specimens were unloaded completely before being subjected to subsequent loadings. The stresses in the axial and circumferential directions for pressure loading were calculated using thin- walled cylinder equations. Note that the sequence of loading is not the same in all

Table 1. Outline of tests on Graphitite-G thin-walled cylindrical specimens

Specimen Load no. type

1 2 : 1 biaxial 0 : 1 axial 1 : 0 circumferential 0 : 1 axial 2 : 1 biaxial

2 2 : 1 biaxial 200 2004 1002 1 1 : 0 circumferential 190 1904 0 2 0 : 1 axial 1000 0 1877 2

3 2 : 1 biaxial 200 2004 1002 2 0 : 1 axial 1000 0 1877 2 1 : 0 circumferential 190 1904 0 2



Maximum internal pressure

(psi)

160

180

213

Maximum Maximum stress axial load Circumferential Axial Number of

(lb) (psi) (psi) cycles

1603 802 1 900 0 1689 1

1804 0 2 1000 0 1877 2

2134 1067

THE BEHAVIOR OF GRAPHITE UNDER RIAXIAI. TENSION 39

cases. Specimen no. 1 was subjected to a maximum internal pressure of 160 psi, while the maximum internal pressure was 200 psi in a11 other cases. In addition, specimen no. 1 was pressurized to failure under combined stress conditions as a final step. The other specimens were not loaded to Sailure.

The strains measured at the gage locations along the lengths and around the circumference of specimen nos. 2 and 3 were carefully examined for each of the three types of loading, that is, internal pressure, axial load, and internal pressure with no net axial load. To facilitate this examination, strain distributions were plotted for various load levels. These plots showed that the peak strains which occurred in the t,ransition regions at the ends of the gage section diminished rapidly with distance toward the center of the specimen. Thus, the central regions were not affected by the discontinuity stresses. There were small differences in the strain values measured around the circumferences at the centers of the specimens. As an example of the results obtained, the strains for specimen no. 3 are shown in Fig. 4 for the 200 psi internal pressure loading. The strain values fi.om gages along a single generator are shown as circles. The strains for other gage locations around the circumference of the specimen are denoted by squares.

Twenty-four tensile specimens were used to obtain stress vs. longitudinal strain and stress vs. lateral strain data. Eight with-

Fig. 4. Strain distribution in Graphitite-G tube No. 3 under 200 psi internal pressure.

grain and eight against-grain specimens were machined from P-in.-diam rod stock, while eight additional against-grain specimens were machined from the tube stock, which had a 2 in. inside diameter and a 23 in. outside diameter. The latter stock was also used to make end sleeves for the thin-walled cylindrical specimens. A modified ASTM specimen design [l 11 was used in each case. Figure 5 shows the design of the with-grain specimen; the gage section was 0.310 in. in diameter and $in. in length. ‘I’he design of the against-grain specimen is shown in Fig. 6, where the specimen is shown super- posed on the rod stock cross section. ,411 of the against-grain specimens had a 0~125in.- diam, &in-long gage section. In the case of the specimens from tube stock, cylindrical rods and end caps, as shown in Fig. i’(A),

LENGTH

Fig. 5. With-grain tensile specimen

DIMENSIONS IN INCHES

Fig. 6. Against-grain tensile specimen.

CAR-Voi.11No.1-D


Fig. 7. Details of against-grain tensile specimen from tube stock.

were made and assembled to form a composite structure. This structure was then machined to the final configuration illustrated in Fig.

7(B). Each against-grain specimen was instru-

mented with four micro-measurement type ~A-13-05OAH-120 strain gages with a 0*050-in. gage length. Two of the gages were oriented in the axial direction and two were oriented in the circumferential direction. The two axial gages were located 180-deg apart. The circumferential gages were mounted to form T configurations with the axial gages. Each with-grain specimen was similarly instrumented with four type CG-111 Budd ~etalfilm strain gages having a 0*063-in. gage length.

Since the parameters, or constants, in the constitutive equations cannot be determined from monotonic loading alone, the specimens were generally cycled twice between zero and a constant maximum stress level before being loaded to failure. Two of the with-grain specimens and one against-grain specimen were loaded to failure without cycling, while one with-grain specimen was cycled between zero and 3600 psi prior to being loaded to failure. The remaining five with-g-rain snecimens were cycled between

zero and 1800 psi as were all of the against- grain specimens, except the one just mentioned.

A typical stress vs. longitudinal strain diagram is shown in Fig. 8. This diagram is for an against-grain specimen that was made from rod stock and cycled twice before loading to failure. The envelope for all stress- strain curves (ignoring the unloading and reloading, or cyclic, portions), the mean of these curves, and the fracture points for the against-grain specimens are shown in Fig. 9. The circled fracture point is that for the specimen which was failed without cycling.

The corresponding stress-strain curve envelope and mean curve as well as the fracture points for the eight with-grain specimens are shown in Fig. 10. The circled fracture points are those for the two specimens which were loaded, to failure without cycling, and the square designates the fracture point for the specimen which was cycled between zero and 3600 psi. These figures show that the scatter in data is not large so far as graphite is concerned and that cyciing had no apparent influence on failure.

500

q, STRAIN (%)

Fig. 8. Stress vs. longitudinal strain diagram for against-grain specimen.

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL TENSION

4000

3000

“, 2000

& t?

to00

0 0 01 02 0.3 04 0.5

cL, STRAIN (-7~)

Fig. 9. Against-grain tensile data.

c t

I I 0.4 0.2 C

EL ,STRAIN (%)

I 1.3 II

0.4

Fig. 10. With-grain tensile data.

4. TEST RESULTS, GRAPHITITE-G MATERIAL

The stress-strain curves from the against- grain and with-grain uniaxial specimens were used to determine the parameters in equation (9), (lo), and (11). These include the elastic moduli, the material constants which characterize the plastic deformations, and the constant c. Lateral-to-longitudinal strain ratio, or total strain ratio, data were also obtained. The total strain ratios were then decomposed into elastic and plastic components through the use of the following

equation [12]:

51

(27)

where

/Z = total, or apparent, strain ratio, E = longitudinal strain (= E’ + cl’), v = Poisson’s ratio, strain ratio in the limit

of zero stress, or strain, p” = plastic strain ratio.

The elastic modulus, E(=E,, =E.J, and plastic constant, B (=B,,), as determined from the initial loading curves, are listed in Table 2 for each of the 15 against-grain specimens that were cyclically loaded. The method used for making these determinations is described in Ref. [8]. Also given are averages of the E and cB values which were determined from the first unloading, first reloading, and second unloading portions of the cyclic curves (see Fig. 8). The average value of E is 1.08 X 106 psi for initial loading, while this value as determined from the unloading and reloading curves is 1.01 X 10” psi. Thus, the two average values are in good agreement. The average value for B is 272 X 10-r” psi-“, and that for cB is 67 X lO_‘” psi-“, giving

a value for c of 0.25 which differs significantly from the value of one-half which was given by Jenkins [4].

The corresponding results for the six with- grain specimens that were cyclically loaded are given in Table 3. The results from the two specimens which were loaded to failure without being cycled are not included. In the case of these data, the average value for E (= Etltl), as determined from the initial loading curves, is 1.86 X 10” psi, and that from the unloading and reloading curves is l-96 X 106 psi. The latter is somewhat higher than the former, while in the case of the against-grain specimens the reverse is true. However, the agreement between the two values is again good. The average H(=N:I:i) value is 73 X 10~” psi?, and the average value for cB is 26 X lo-‘” psiCz. Thus, the value

W. L. GREENSTREET, G. T. YAHR and R. S. VALACHOVIC

Table 2. Graphitite-G against-grain tensile data

Initial loading Unloading and reloading

Specimen E B E ave (cBLwe Stock no. ( lo6 psi) (10-i’ psiP) ( lo6 psi) ( 1O-‘2 psi-*) material

RI 1.13 277 1.01 71 R2 1.02 288 0.96 78 R3 1.07 334 0.93 76 R4” 0.92 64 R5 0.98 364 0.93 85 R6 0.99 298 0.95 80 R7 1.04 284 0.95 75 R8 1.14 268 1.02 60

Rod stock: 2-in-diam

Average: 1.05 302 0.96 74

Tl 1.05 290 0.97 54 T2 1.11 281 1.02 63 T3 1.10 268 1.04 72 T4 1.14 225 1.09 61 T5 1.15 230 1.10 55 T6 1.10 220 1.08 63 T7 1.16 175 1.13 51

Average: 1.11 241 1.06 60

Tube stock: 2-in.-ID X

2Qin.-OD

Average for combination of rod and tube specimens:

1.08 272 1.01 67

*The initial loading portion of the curve obtained was not suitable for evaluating the constants. The results from this specimen are not included in Fig. 9.

Table 3. Graphitite-G with-grain tensile data

Initial Unloading and reloading

Specimen E

no. (IO” psi)

Rl 1.93 R2 1.87 R3 1.78 R4 1.88 R5 1.90 R6 1.80

;; 76 73 87 72

E ave W&e Stock ( lo6 psi) ( lo-i2 psim2) material

1.84 17 2.04 31 1.88 17 2.09 39 1.97 25 1.91 28

Rod stock: 2-in.-diam

Average: 1.86 73 1.96 26

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL. TENSION

for c is 0.36, which again is significantly different from one-half. Because c is assumed to be a constant that is independent of the orientation of the specimen, the average of the two values, or 0.30, will be used in the analysis described in this paper.

The average total strain-ratio curve, which is designated by p,, in Fig. 11, was obtained from seven with-grain specimens. Since the elastic strain ratio is equal to the total strain ratio in the limit of zero longitudinal strain, vQ1 is represented by the straight line parallel to the l~~l~gitudinal strain axis in this figure (y:t1 = 0.126). Equation (27) was used to cal- culate the curve for the plastic strain ratio, &if. In this case, @, is essentially independent of the longitudinal strain and it is negative in the range shown. However, the apparent strain-ratio data lack precision in the small strain range shown, and additional inaccura- cies are introduced in the decomposition process. Hence, the value of @, can be assumed constant without introducing significant error. By giving greater weight to the values at larger longitudinal strains, a value of -0.08 was selected for the purposes here.

The total strain ratios, pip, were determined from three against-grain specimens, and values for v12 and pf;, were obtained in the same manner as the vS1 and &I values dis- cussed above. These are listed in Table 4 where it can be seen that there are significant differences between the numbers for the three specimens. However, the plastic strain ratio is negative in each case. The average

O.z r-- .--“- ._~ .__--

I 7

-0.1

-0.2 /____.-_.--..-_ / / / ~~

0 0.05 0.10 0.15 0.20 0.25 0.30 ei. LONGITUDINAL STRAIN f%l

Fig. Il. Average strain-ratio data for seven Graphitit.e-G with-grain tensile specimens.

Table 4. Graphit.ite-G strain- ratio data from against-grain

tensile specimens

Specimen no.

R7 O-20 -0. 13 R8 0.26 -0.28 TS 0.14 -Q*O5

value for v12 from the three specimens is 0.20, while that for prz is -0~1.5.

Since the strain ratios needed in equations (23) through (26) are v13 and P;~, these quantities were calculated from the above values for v:$, and @&. Equations (21) give the following relationship between the elastic strain ratios:

E 1,

-__._u 13 - & %l> (28)

and equation (22) give the following relationship between the plastic strain ratios:

(29)

The values thus determined for v~:~ and &\ are WO7 and -@03, respectively.

Although sKL and CL%:. values, as required in equations (23)-(26), were determined from the tensile specimens, variations in material properties preclude their use in making definitive comparisons between calculated and experimental results for the cylindrical specimens under internal pressure loading. The strain-ratio values and the value for c will be used, however, The uniaxial tests conducted on each cylindrical specimen provided means whereby the masking effects due to material variability could be overcome. But it was necessary to determine the needed data in such a way that the effects of prior loading histories did not inHuence the values obtained.


After loading a specimen in a given direction in stress space,? the stress-strain response obtained upon loading in a different direction is not the same as loading a virgin specimen in the second direction. However, the model of graphite behavior given in Refs. [l] and [2] predicts that, once the specimen has been loaded to a given stress level in the new direction, unloading and reloading curves subsequently obtained by cycling between

where u& is the stress at the unloading point, are the same as the unloading and reloading curves that would have been obtained using a virgin specimen. Therefore, A and CB

values for each of the cylindrical specimens were obtained from cyclic unloading and reloading curves produced by the applica- tion and release of an axial force and by cyclic pressurization with no net end load. The B values were then calculated using the value for c from the tensile tests. The average A and B values obtained in this manner are

tin this case we are concerned with a two- dimensional space in which the axes are labeled by crz and v~, that is, the axial and circumferential stress components, respectively, and we further restrict ourselves to loadings for which the ratio of a2 to a3 remains constant during a given loading.

listed for each of the five cylindrical specimens in Table 5. The overall averages for the cylindrical and tensile specimens are given at the bottom of the table.

A comparison of the results in Table 5 with those of Tables 2 and 3 shows that, in general, the A and B values are larger for the tensile specimens. Also, Table 2 indicates that the B values for specimens made from rod stock are higher than those from the tube stock, but the value for c is essentially the same for the two cases. Finally, it may be seen that the variations in B values for the cylindrical specimens are less than those for the tensile specimens.

Since strain ratios for metals and other materials are positive, a question naturally arises regarding the validity of the stress- plastic strain relations for the case where negative plastic-strain ratios exist. The validity in the case at hand is determined through an examination of the plastic potential function, which in the mathematical theory of plasticity must represent a convex surface in stress space[l3,14]. Therefore, a statement of the requirements for convexity of surfaces given by equation (20) is necessary.

Convexity is assured by requiring that the matrix of coefficients,

bKL/k

be positive definite. The nonzero coefficients

Table 5. Data from cylindrical specimens

Specimen no.

An A B11 B3 (10e6 psi-‘) (10m6 Tsi-I) (lo-‘* psiP) (lo-l2 psi+)

1 2 3 4 5

Averages: Cylinder data Tensile data

0.73 0.56 123 42 0.78 0.47 133 39 0.80 0.58 148 50 0.76 0.46 136 40 0.77 0.55 135 51

0.78 0.52 138 45 0.92 0.54 272 73

THE BEHAVIOR OF GRAPHITE UNDER BIAXIAL TENSIOK 55

in this matrix and the equivalences in terms of measured quantities are given by equation (22). By imposing the positive definite requirement, the following inequalities are obtained [15, 161:

B,, > 0. & > 0, H14 > 0, (30) -1 < /..LLz< 1.

( 1 - /qz. 1 ’ %$/4~.

These inequalities show that the strain ratios can be positive or negative so long as they remain within the bounds given. The B values are greater than zero in all cases. Further, it may be seen that the fourth and fifth inequalities are satisfied. Thus, the surfaces described by equation (20) are convex.

5. COMPARISONS OF CALCULATED AND EXPERIMENTAL RESULTS

The axial and circumferential strains for cylindrical specimens nos. 1 and 2 under the initial internal pressure loading cycles were calculated using equations (23)-(26), the constants in Table 5, and the strain ratios and c value from the tests on tensile specimens. The experimentally determined curves for these specimens are shown in Figs. 12 and 13 where the points calculated from the equations are indicated by the open circles. The experimental and theoretical results shown are in excellent agreement. Similar agreement was found for the remaining specimens.

Figure 14 shows the comparisons between theoretical and experimental results re- presenting the averages from specimens nos. 2 through 5 when the A and B values obtained from these cylindrical specimens are used. Again, the figure shows that the results are in excellent agreement.

The results of the studies given in this report demonstrate the need for determining uniaxial data directly from the specimens

0 0.05 0.10 o.r5 0.20

STRAIN vo)

Fig. 12. Comparisons of measured curves with calculated results for Graphitite-G specimen no. I.

2000

500

0

-CIRCUMFERENTlA

- CALCULATED

0 0.05 0.10 0.!5 0 20

STRAIN Phi

Fig. 13. Comparisons of measured curves with calculated results for Graphitite-G specimen no. 2.

tested under combined states of stress. However, investigations using other graphites have shown that very good agreement between calculated and experimental results for graphites exhibiting small variations in mechanical properties are obtained when data from tensile specimens alone are used in the analyses. Thus, for some graphites,

W. L. GREENSTREET, G. Y. YAHR and R. S. VALACHOVIC 56

2000

1500

::

t-z g 8000

b

500

t

0 k 0 0.05 0.10 0.t5 0.20

1 STRAIN (%)

Fig. 14. Comparisons of averaged measured curves and averaged calculated results for

Graphitite-G specimen nos. 2-5.

the precautions taken here in obtaining materials properties data are not necessary, and the equations may be used with con- fidence for practical applications when stress- strain data from the uniaxial specimens alone are obtained.

To illustrate this point, a comparison between calculated and experimental results for a specimen made from a specialty graphite are shown in Fig. 15. The parameters used in the constitutive equations

1000

2 600 a

9 ks ‘, 400

200 -MEASURED

STRAIN i%)

Fig. 15. Comparison of measured curves with calculated results for a specialty graphite.

were:

E,, = 127 X IO6 psi B,, = 99 X IO-12 psi+ Ea3 = 2.37 X 106 psi Bz3 = 36 X 10-12 psi-” V 13 = 046 /_l$ = -0.10

c = 0.25

where all of the values were determined directly from tests on uniaxial specimens. Again, the agreement between theory and experiment is very good.

6. CONCLUSIONS It is shown that nonlinear stress-strain

behavior of graphite under combined stress conditions at room temperature can be accurately described by a small-deformation elastic-plastic continuum theory. Results calculated through the use of specia1 constitutive equations appropriate to the des- scription of graphite behavior were shown to be in excellent agreement with experimental results corresponding to initial Ioading and to unloading under combined states of stress produced by internal pressure in thin- walled cylindrical specimens. Further, the calculated residual strains upon unloading were very close to those determined experimentally.

In the examination of a theory, such as that described, it is essential that experi- ments be designed to overcome problems associated with the accurate determinat.ion of material constants and to minimize masking effects due to material variability. Because of the variability in mechanical behavior of the primary material used, parameters for use in the constitutive equations in this investigation were determined using results from tensile specimens along with uniaxial data obtained from the thin-walled cylindrical specimens. The results derived establish, in part, the validity of the theory. Additional investigations to examine this theory should be designed to study the basic postulates since the validity has been established, as shown here, in what may be considered a gross sense.

THE BEHAVIOR OF GRAPHITE UNDER BI.4XIAL TENSION .55

Acknowledgements-The authors gratefully ack- nowledge the assistance of H. A. MacColl in the data reduction, and J. M. Chapman for his help in 10. instrumenting and testing the specimens. They- also thank C. E. Pugh for his careful review and helpful criticism of this work. 11

1.

2.

3.

4. 5.

6.

7. 8.

9.

REFERENCES

Greenstreet IV. L.. Ph.D. dissertation, Yale University (1968). Greenstreet W. L. and Phillips A., Acta Mechanica, to be published. 12 Greenstreet, W. L. and Phillips, A., Proc. First Int. Co?/. on Structural Mechanics in Reactor Technology, to be published. 13. Jenkins, G. M., Br.J. Appl. Phys. 13,30 (19fi2). 14. .Jenkins, G. M. and Williamson, G. K.,J. Appl. Phys. 14,2837 (1963).

Theory of Elasticity, 4th Ed., Dover, New York (1944). Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity, Oxford University Press, London (1961). ‘Standard Methods of Tension ‘I‘estinq of Carbon-Graphite Mechanical Mate*‘ials,’ ASTM Designation: C565-71, pp. 4%-499 in 1971 dnnual Book of ASTiZl Standards, Part 13, American Society for Testing and Materials, Philadelphia (1971). Shanley. F. R., Weight-Strength .4ualysi.\ o/ Aircraft Structures, 2nd Ed.. Dovrl-, New York (1960). Prager, W.,j. A#/. Phys. 20,235 (1949). Drucker, D. C., Proc. First I’S. National C;ongre.\.\ of Applied ;Mwhrcnic.c, pp. 387-491, New York (1952).

Rappeneau, J.. Jouquet, C., and Guimard, 15. Stein, F. M., Introduction to Matrzcas mind M., Carbon 3,407 (1966). Determinants, PP. 100-101, Wadsworth. Seldin, E. J., Carbon 4, 177 (1966). Belmont, California (1967). Greenstreet, I&‘. L., Smith, J. E., and Yahr, 16. Born, M. and Huang, K., Dynamical Th,eor)j of G. T., Carbon 7, I5 (1969). Crystal Lattices, p. 141, Clarendon Press, Love, A. E. H., A Treatise on the Mathematical Oxford (1954).

the behavior of graphite under biaxial …academic.csuohio.edu/duffy_s/temp/doe caes program/stress...

Documents